The Queen may be the head of state, but she does not run the Government.

Nothing there becomes law without her signature. Even the Prime Minister doesn't have that power.

Republics such as Israel, following the Parliamentary system often appoint their presidents rather than elect them, but nobody would suggest that they are not representative democracies...

That, like our presidency here in the U.S., would be an indirect democracy which is clearly a type of democracy. The U.K. uses a hereditary monarchy to choose their head of state, and there is nothing democratic about it.

The Queen’s power to refuse to sign legislation is there in theory, but then again, since the 1688 Bill of Rights was concocted, monarchs only sit upon the throne by the permission of Parliament, so if the Queen tried to rule against the express will of Parliament, Parliament could declare her deposed and put someone else on the throne instead, or even abolish the monarchy entirely.
At least in Britain, most members of the cabinet are elected public officials, last time I checked, the US President appoints people who are not members of any legislative body.
Not saying that there is anything wrong with this though, there is such a thing as too much democracy....